I need to prove that any rational number a/b can be expressed as the sum of 4 squares of rational numbers
I have tried something similar to the proof for integers, but can't get it to work. I have tried an algebraic proof, but I keep running into the issie of potential irrational denominators
You are taking $\frac{a}{b}$ with integers $a,b > 0.$ Thus $ab$ is a positive integer.
Let $$ w^2 + x^2 + y^2 + z^2 = ab $$
Then $$ \left( \frac{w}{b} \right)^2 + \left( \frac{x}{b} \right)^2 + \left( \frac{y}{b} \right)^2 + \left( \frac{z}{b} \right)^2 = \frac{a}{b}$$